Question

1 multiple choice 20 points 5.2a which end behavior diagram describes the function, $f(x) = -5x^4 - x^2 + x + 3$?

Explanation:

Step1: Identify the leading term

The leading term of the polynomial \( f(x) = -5x^4 - x^2 + x + 3 \) is \( -5x^4 \). The degree of the polynomial is 4 (even) and the leading coefficient is -5 (negative).

Step2: Determine end behavior for even degree

For a polynomial with even degree:

• If the leading coefficient is positive, as \( x \to \pm\infty \), \( f(x) \to +\infty \).
• If the leading coefficient is negative, as \( x \to \pm\infty \), \( f(x) \to -\infty \).

Here, degree is 4 (even) and leading coefficient is -5 (negative). So as \( x \to \infty \), \( f(x) \to -\infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \). This corresponds to the end behavior diagram where both ends point downwards (the first option with two downward arrows).

Answer:

The end behavior diagram with both ends pointing downwards (the diagram with two downward - facing arrows)